Knowledge of the dynamic behavior of a system is beneficial in many science and engineering applications including, for example, electronic systems, electrical systems, acoustic systems, hydrodynamic systems, biosensors, biomimetics and nanotechnology. The dynamic behavior of a system is defined by the relationship between an input or stimulus to a system and the output or response of the system to that input. Under analysis of a system in the frequency domain, the relationship between the stimulus and the response is commonly referred to as a system's transfer function.
In order to determine the transfer function of a system it is necessary to calculate the magnitude of the response which we here refer to as the “gain” of the system and the phase shift between the stimulus and the response. Generally, the gain and phase shift are based on measurements of an input waveform applied to the system and the resulting response waveform of the system. However, in practice there are limitations to obtaining accurate stimulus and response measurements.
One difficulty in making such measurements arises when coupling a stimulus to a system and when coupling response measurement instrumentation to the system as the coupling equipment introduces additional signals such as noise. Errors in transfer function calculation occur because the response measurements will include the additional signals superimposed such that the resultant transfer function is not an accurate reflection of the dynamic behavior of the system.
In order to reduce the effect associated with interface equipment a first set of interface elements may be used to apply the stimulus to the system and a second set of interface elements may be used to measure the response. However, in order to reduce the effect of the interface equipment, the first and second interface elements must have very high sensitivity capabilities which adds substantially to the cost of the equipment.
A further difficulty in making accurate measurements to determine a transfer function of a system arises due to the presence of background signals which, although undesirable, are unavoidable. These background signals are included in the measurement of the response waveform and thus a true response waveform is not sampled. For example, background signals can arise from nonlinear characteristics of a system which distort response waveforms. Even when a stimulus of very small amplitude is applied in an attempt to reduce errors arising from nonlinearities in the response system other background components may be present in the responses. Such other background components may include, for example, transient responses of a system, distortion generated by the acquisition instrumentation, thermal fluctuations and other dynamic fluctuations generated by the system and noise generated by the instrumentation.
U.S. Pat. No. 4,214,203 to Coster et al relates to a method of measurement of electrical impedance by measuring the gain and phase shift of sinusoidal waveforms. The method described is limited to electrical impedance measurement and, in particular, to a stimulus having a sinusoidal waveform of a single frequency. In order to obtain the transfer function of a system by applying the method of U.S. Pat. No. 4,214,203 the stimulus must be applied at a first frequency, then at a second frequency and so on over a range of frequencies to measure a spectrum of impedances and to thereby obtain the transfer function of the system to a sinusoidal stimulus.
U.S. Pat. Nos. 6,556,001 and 6,885,960 relate to a method of acquiring impedance spectra of non-stationary systems dynamically. Generally, U.S. Pat. Nos. 6,556,001 and 6,885,960 relate to the concept of using complex waveforms consisting of a combination of sinusoidal waveforms of various amplitudes and frequencies used to acquire impedance spectra of systems more rapidly. In particular, U.S. Pat. Nos. 6,556,001 and 6,885,960 use a stimulus of structured noise wherein the structured noise is achieved by the superposition of a finite number (between 5 and 50) of sinusoidal oscillations. Such a stimulus may cover a wide bandwidth and a range of amplitudes. By using such waveforms as a stimulus to a system together with response analysis, involving Fourier de-convolution techniques, impedance spectroscopy data can be acquired more rapidly. However, the very large number of frequency components to be extracted from the responses fundamentally limits the resolution of relative phase and amplitudes of the individual sinusoidal components. While this method provides useful information about the impedance of a system at a number of different frequencies simultaneously, the method is not applicable to the measurement of a transfer function for the case of arbitrary waveforms and cannot provide very precise measurement of the phase of either the waveform component in the responses independently of the amplitude of background components or of the individual sinusoidal components in the responses to the stimulus when the latter is comprised of a combination of sinusoidal waveforms.
A common method used to measure the phase difference between, for example, two sinusoidal waveforms is to measure the time delay between zero crossings of a stimulus waveform and a response waveform. This is essentially an analogue method, even when the timing between zero crossings is performed digitally. However, in practice the accuracy of this method is limited by noise being superimposed onto the response waveform which results in a plurality of erroneous zero crossings. Further limitations arise due to DC offsets in the response which usually lead to a shift in the zero-crossing point. For complex waveforms, such as waveforms composed of multiple sinusoidal components, there may be several zero crossings within each period, which would lead to an ambiguity in the phase difference measured between the stimulus and the response. It is inherently difficult to measure zero crossings due to finite thresholds and hysteresis in real measurement devices.
Traditional methods of determining relative phase typically are not suitable for use with arbitrary waveform stimuli including arbitrary waveforms constructed from a combination of sinusoidal signals using the mathematical method of Fourier synthesis.
Any discussion of documents, acts, materials, devices, articles or the like which has been included in the present specification is solely for the purpose of providing a context for the present invention. It is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present invention as it existed before the priority date of each claim of this application.
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